Answer
$\frac{16}{25}=0.64$
Work Step by Step
Let's define the events:
$A_1$: "send her an e-mail"
$A_2$: "leave a message on her phone"
$B$: "she responded within three hours"
$P(A_1)+P(A_2)=1$ but $P(A_1)=2P(A_2)$
$2P(A_2)+P(A_2)=1$
$P(A_2)=\frac{1}{3}$ and $P(A_1)=\frac{2}{3}$
$P(B~|~A_1)=0.80$
$P(B~|~A_2)=0.90$
$P(A_1~|~B)=\frac{P(B~|~A_1)P(A_1)}{P(B~|~A_1)P(A_1)+P(B~|~A_2)P(A_2)}$
$P(A_1~|~B)=\frac{0.80\times\frac{2}{3}}{0.80\times\frac{2}{3}+0.90\times\frac{1}{3}}=\frac{\frac{1.6}{3}}{\frac{1.6+0.9}{3}}=\frac{16}{25}=0.64$
